On fractional $(p,q)$-calculus

Abstract

In this paper, the new concepts of (p, q)-difference operators are introduced. The properties of fractional (p, q)-calculus in the sense of a (p, q)-difference operator are introduced and developed.

Multiple Positive Solutions for a Class of Boundary Value Problem of Fractionalp,q-Difference Equations underp,q-Integral Boundary Conditions

“…Lemma 3 (see [15]). For α, β ≥ 0 and 0 < q < p ≤ 1, (p, q)-integral and (p, q)-difference operators have the following properties:…”

“…Definition 1 (see [15]). For 0 < q < p ≤ 1 and f: [0, T] ⟶ R, we define the (p, q)-difference of f as…”

“…Definition 2 (see [15]). For N − 1 < α < N, 0 < q < p ≤ 1, and f: I T p,q ⟶ R, the fractional (p, q)-difference is defined by…”

“…(ii) D p,q [αf(t)] � αD p,q f(t), for α ∈ R Lemma 2 (see [15]). For 0 < q < p ≤ 1, α ≥ 1, and a ∈ R, we have…”

“…On the contrary, the research of fractional calculus in discrete settings was initiated in [8,11,14]. In 2020, Soontharanonl and Sitthiwirattham [15] introduced the fractional (p, q)-calculus, which has been found in a wide range of applications in many fields such as concrete mathematical models of quantum mechanics and fluid mechanics [7,13,15].…”